# Probe 6: Bad Controls Test (Angrist & Pischke)

If Z predicts a control variable, that control is a *mediator* (part of the
demographic mechanism), not a *confounder*. Including mediators in the CA
regression biases the demographic coefficient toward zero.

## Z → Potential Mediator Regressions

| Dep. Variable | Sample | Z₁ Coef | Z₁ SE | Z₁ p-val | R² | N | Countries |
|---------------|--------|---------|-------|----------|----|---|-----------|
| health_exp_gdp | 69-country | 8.059** | 3.927 | 0.0403 | 0.386 | 1604 | 67 |
| health_exp_gdp | 140-country | -1.440 | 1.890 | 0.4462 | 0.304 | 3309 | 139 |
| life_expectancy | 69-country | 21.450*** | 7.781 | 0.0059 | 0.470 | 2992 | 68 |
| life_expectancy | 140-country | 39.565*** | 5.445 | 0.0000 | 0.613 | 6160 | 140 |
| expected_growth | 69-country | 20.281*** | 5.222 | 0.0001 | 0.136 | 3039 | 69 |
| expected_growth | 140-country | 18.030*** | 4.769 | 0.0002 | 0.037 | 5944 | 141 |
| log_rel_opw | 69-country | 2.082*** | 0.543 | 0.0001 | 0.582 | 2740 | 69 |
| log_rel_opw | 140-country | 1.022** | 0.448 | 0.0224 | 0.451 | 5409 | 141 |

## Interpretation

- Variables where Z₁ is significant (p < 0.05) are likely **mediators**.
- Including mediators in the CA regression removes the demographic *channel*,
  not confounding — this is a "bad control" problem.
- If life_expectancy and health_exp are mediators, the parsimonious spec
  (4 controls, Z₁ p=0.017) is the appropriate specification.
